In the 1960s, it wasn't always possible to detect rockets going off, there was a system of satellites in the that decade but it was geared towards detecting Soviet nukes in the boost phase and thus focused entirely on the northern hemisphere:
Though I hesitate to spend too much effort (it's far from a serious movie), Dr. Evil would have had to have lifted all the materials into orbit from South America or possibly Australia. There aren't very many nations that would be friendly to an international terrorist threatening western governments in either of those places. So he may have novel spaceflight technology not reliant on rocket technology (anti-grav, etc).
You are correct to assume that whatever the specific details, he would be moving many thousands of tons of equipment and many hundreds of people through this clandestine spaceport to be able to build a moon base. To construct it he would have needed to put dozens of people on the moon (and if memory serves, the only nation to do so managed to send exactly 1 dozen people to the moon... dozens is inconceivable).
It would have had a multi-trillion dollar price tag (inflation-adjusted), required a small city of workers, and been the most impressive engineering feat ever accomplished by mankind.
Conclusion: mildly implausible.